De Bruijn–Newman constant

The de Bruijn–Newman constant, denoted by <math>\Lambda</math> and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function <math>H(\lambda,z)</math>, where <math>\lambda</math> is a real parameter and <math>z</math> is a complex variable. More precisely,

:<math>H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2 \Phi(u) \cos (z u) \, du</math>,

where <math>\Phi</math> is the super-exponentially decaying function

:<math>\Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u</math>

and <math>\Lambda</math> is the unique real number with the property that <math>H</math> has only real zeros if and only if <math>\lambda\geq \Lambda</math>.

The constant is closely connected to the Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the statement that <math>\Lambda\leq 0</math>. Brad Rodgers and Terence Tao proved that <math>\Lambda\geq 0</math>, so the Riemann hypothesis is equivalent to <math>\Lambda=0</math>. A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.

History

De Bruijn showed in 1950 that <math>H</math> has only real zeros if <math>\lambda\geq 1/2</math>, and moreover, that if <math>H</math> has only real zeros for some <math>\lambda</math>, <math>H</math> also has only real zeros if <math>\lambda</math> is replaced by any larger value. Newman proved in 1976 the existence of a constant <math>\Lambda</math> for which the "if and only if" claim holds; and this then implies that <math>\Lambda</math> is unique. Newman also conjectured that <math>\Lambda\geq 0</math>, which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

Heat-flow interpretation

The family <math>H_\lambda</math> may be viewed as a deformation of the Riemann xi function under a heat-type equation. At <math>\lambda=0</math>, the function <math>H_0</math> is essentially the Riemann xi function, written as an even Fourier transform. Varying <math>\lambda</math> multiplies the Fourier-side kernel by <math>e^{\lambda u^2}</math>. Differentiating under the integral sign gives

:<math>{\partial H_\lambda\over\partial \lambda}=-{\partial^2 H_\lambda\over\partial z^2},</math>

so that increasing <math>\lambda</math> evolves <math>H_\lambda</math> by the backward heat equation in the variable <math>z</math>.

Upper bounds

De Bruijn's upper bound of <math>\Lambda\le 1/2</math> was not improved until 2008, when Ki, Kim and Lee proved <math>\Lambda< 1/2</math>, making the inequality strict.

In December 2018, the 15th Polymath project improved the bound to <math>\Lambda\leq 0.22</math>. A manuscript of the Polymath work was submitted to arXiv in late April 2019, and was published in the journal Research In the Mathematical Sciences in August 2019.

This bound was further slightly improved in April 2020 by Platt and Trudgian to <math>\Lambda\leq 0.2</math>.

Historical bounds

{| class="wikitable"

|+ Historical lower bounds

!Year !! Lower bound on Λ

|1987 ||−50

|1990 ||−5

|1991

|−0.0991

|1993 ||−5.895

|2000 ||−2.7

|2011 ||−1.1

|2018 || 0